An ink for a fading sheet is set for its middle age

A sheet of brightened paper glows: its brightener absorbs ultraviolet and re-emits blue, and the sheet returns more visible light than a plain paper of the same base. A pale tint printed on it glows only where its paper is bare, because the ink’s vehicle absorbs the ultraviolet, and adapting to a brightened sheet leaves the glow found that a reader adapted to the sheet still sees the tint’s hue turned by the mismatch. A light ink can cancel the glow at one strength then printed each pigment diluted, over more of the paper, and found one strength per pigment at which the residue vanished.

One light ink serves three papers within two degrees carried those inks across the census’s three brightened sheets. Each pigment’s cancelling strength fell as the sheet’s glow rose, along nearly straight lines in the share of the sheet’s luminance that is glow, and an ink set for the middle sheet erred on either side by under two degrees. Its closing section turned to a sheet’s life. The brightener is being used up found that a brightener breaks as it works, so that a sheet’s whiteness has a half-life; a printed page read for a year is not on the paper it was printed on.

The prediction was that as the premium sheet’s brightener bleaches, every pigment’s cancelling strength rises back along the line — reaching the office sheet’s strength where the bleached glow reaches the office sheet’s — and that the fresh premium ink stays within two degrees until half the brightener is gone. The alternative named was a strength that does not track the glow on a bleached sheet, which would mean the brightener’s loading and spectrum matter separately and an ink set for a sheet’s glow is set for one day of its life.

The line holds, and the middle ink wins #

With a quarter, a half and three quarters of its brightener bleached, the premium sheet’s glow falls from 0.84 per cent of its luminance to 0.77, 0.65 and 0.44, and every pigment’s cancelling strength rises at each step — the orange’s from 0.509 to 0.520, 0.541 and 0.599, the yellow’s from 0.641 to 0.749 — within 0.015 of the census line at each glow and never more than 0.003 above it. Printed with the fresh premium sheet’s inks, the bleaching sheet’s worst residue from five to twenty per cent is 1.75, 1.80 and 2.35 degrees. Printed with the office sheet’s inks, it is 1.39, 1.13 and 1.37.

• The prediction holds on both counts. The strength walks back down the line, and the fresh ink holds within two degrees exactly until half the brightener is gone.
• The alternative fails. Loading and spectrum do not need to be read separately; the glow carries nearly all of it.
• What the prediction did not ask is the useful part. The ink that serves the sheet’s whole life is not the fresh sheet’s own.

The three curves start in a familiar order. On the fresh sheet the full-strength ink leaves 12.6 degrees, the premium sheet’s own light ink 1.71 and the office sheet’s 1.80: the premium ink is set for this sheet and errs only along the ramp, the office ink is set for a dimmer sheet and falls short. As the brightener bleaches, the full-strength residue falls with the glow, the premium ink’s error rises slowly to half bleached and then quickly, and the office ink’s falls at the first step and stays under one and a half degrees.

Read as a choice of ink for a printed piece that will be read for its whole life, the office sheet’s ink is the better one from the first quarter of bleach onwards and never worse than 1.8 degrees, where the premium sheet’s own ink reaches 2.35 by the end.

Back along the line #

The bleached premium sheet is a new paper — the census never held it — and its strengths fall on the line drawn through the three papers the census did hold. The orange’s bleached strengths are 0.520, 0.541 and 0.599 against 0.522, 0.554 and 0.610 on the line; the yellow’s 0.656, 0.684 and 0.749 against 0.658, 0.693 and 0.754; the green’s barely move at all, as they barely moved across the census sheets.

That is a prediction from three points tested on three more, and it is the strongest result here. The line was drawn through sheets that differ in brightener loading and in their base paper — the light sheet’s base is a little yellower than the premium’s — and it reads a bleached sheet whose base is the premium’s and whose loading is anywhere from three quarters of the premium’s to a quarter of it. A printer who knows a sheet’s glow, by measuring its fluorescence or its whiteness under an ultraviolet-containing lamp, knows the light ink it wants.

A shade under the line #

The fit is close and not exact, and its departure has a sign. Every bleached sheet’s strength lies within 0.015 of the line, and none is more than 0.003 above it; the largest gap is the orange’s at half bleached, 0.013 below. A bleached premium sheet asks for a slightly weaker ink than a census sheet with the same glow would.

The likeliest reason is the paper beneath the brightener. Bleaching removes brightener and leaves the premium sheet’s base, which is whiter and less yellow than the office or light sheet’s, and a whiter base makes the bare paper a slightly larger share of a pale tint’s light. The census does not separate the two; holding the base fixed while bleaching, as it does here, is what exposes the gap, and the gap is small enough that the line remains the useful reading. Real photo-ageing yellows the base as it bleaches the brightener, and the census line, drawn through yellower bases at lower glows, may already carry part of that.

The fresh ink overshoots as the sheet fades #

Pigment by pigment, the fresh premium ink’s five-per-cent residues move steadily the same way as the brightener goes. The orange’s runs from −0.12 degrees at a quarter bleached to +0.42 at half and +1.45 at three quarters; the cyan’s from +0.04 to −0.70, its full-strength residue’s sign being the other way. Each moves away from where its full-strength residue would be, past nought: the overshoot the office ink showed on the light sheet in the earlier essay. A fading sheet has become a paper with less glow than the ink was set for, and the ink covers more of it than it now needs.

The worst residue, over the whole ramp from five to twenty per cent, is set by the twenty-per-cent tints, which carry the ramp’s own drift. On the fresh sheet the premium ink already errs by 1.71 degrees at twenty per cent on the red, because a light ink set to cancel at ten per cent overshoots at twenty on any sheet; half the brightener’s bleach adds only a tenth of a degree to that, and three quarters adds six tenths.

The middle ink from both sides #

The office sheet’s ink, carried onto the bleaching premium sheet, crosses nought instead of moving away from it. With a quarter bleached, the orange’s five-per-cent residue is −1.39 degrees, still on the full-strength side; with half, −0.74; with three quarters, +0.52, past nought. Every pigment does the same, falling short early and overshooting late, because the bleaching sheet’s glow passes through the office sheet’s — 0.572 per cent — between half and three quarters bleached.

That is the reason it serves the whole life. An ink whose setting the sheet passes through is wrong in one direction before and the other after, and never by much, because the residue changes sign smoothly through its zero; an ink set for one end of the sheet’s life is wrong in one direction for the rest of it, and increasingly. The earlier essay found the same symmetry across papers; here it is across time on one paper.

What there is left to cancel #

The bleach also helps the full-strength ink. Bleached by three quarters, the premium sheet keeps 52 per cent of its glow, and every pigment’s full-strength residue keeps between 0.50 and 0.61 of its size: the red’s from 12.6 degrees to 7.4, the orange’s from 10.0 to 5.6. Drawn against glow, the bleached sheets continue the census sheets’ rise of residue with glow.

So a printer who prints at full strength sees a pale tint’s residue roughly halve over a sheet’s life, and a printer who prints with light inks sees it change sign. Neither stands still, and the light inks’ changes are an order of magnitude smaller: under one and a half degrees with the office inks, against changes of five degrees and more at full strength.

Two clocks on one page #

Squares of four print a pale tint like a 150-line screen and the screening essays before it, among them stochastic dots hold the glow at every tint, measured the residue on one sheet at one moment. The brightener adds a clock to the page: the same printed sheet presents a different paper each month it is read in daylight, and a residue cancelled on the day of printing comes back, turned the other way. Merged spots give the mid-tones half the glow back found that a screen’s geometry decides how much of a tint’s edge lets glow through; nothing in the screen changes as the sheet ages, so the geometry’s share of the residue presumably ages with the glow as the full-strength residue does here — a reading the census has not made.

The reader’s own adaptation is a second clock running on a different time scale. A reader adapts to the sheet in hand within a minute, and the census’s residue is for that fully adapted reader; the brightener’s clock runs over months. Because every residue here is read by a reader adapted to the sheet as it is at that moment, a bleached sheet’s whiteness is discounted as fully as a fresh one’s — what is left is the glow’s mismatch between bare and inked paper, and that is what the ink’s strength sets.

What a whiteness reading would say #

A printer who wants to set an ink by a sheet’s glow needs a way to read the glow, and the obvious instrument already exists. Whiteness is mostly the lamp found that the CIE whiteness formula, read under an instrument lamp with ultraviolet in it, separates a premium sheet from an ordinary one almost entirely through the brightener’s contribution — take the ultraviolet out and most of the scale collapses. That is a weakness in a whiteness figure as a description of a paper’s colour, and a strength here: a whiteness reading with and without the ultraviolet is close to a reading of the glow itself.

The bleached premium sheets give that a concrete use. Their glow falls from 0.84 per cent of luminance to 0.44 over the three stages, and the ink each wants falls on the census line within a hundredth or so; a printer who reads a stock’s whiteness both ways at the mill, and again on a sample aged as the printed piece will be, has the two ends of the line and can set the ink between them. Some paper is brighter than white is the reminder of why the reading has to include the ultraviolet at all: a fluorescent sheet has no reflectance curve, and any measurement made without the light that excites it measures a different paper.

The other route is to print a control. A strip of pale tints in each pigment, printed with the light inks at the margin of a sheet and read for its hue turn as the sheet ages, would show the residue crossing nought at whatever point in the sheet’s life the ink was set for; on this census’s lines, a strip that crosses nought between a half and three quarters of the brightener’s bleach is printed with the right ink.

What a printer should take from this #

Set a light ink for the glow the sheet will have in the middle of its life. Printed on the premium sheet, the office sheet’s inks — set for a glow the premium sheet reaches between half and three quarters bleached — keep every pale tint within 1.4 degrees from the first quarter of bleach to the last, where the premium sheet’s own inks drift to 2.35.

Measure the glow, not the loading. A bleached sheet’s cancelling strength sits on the line the census sheets draw in glow, within a hundredth or so; a sheet’s brightener loading alone, which bleaching changes, does not say where.

Expect the residue to change sign, not to vanish. Any light ink cancels on one day of a brightened sheet’s life. A sheet expected to be read for years should be printed with an ink that cancels in the middle of them.

How the bleached sheets were built #

The premium stock is the census’s: a paper base of level 0.9 and yellowing 0.13 with a brightener at loading 2, under D65 extended to 300 nm. A bleached sheet keeps the base and scales the brightener’s loading by one less the bleached share, and is built by the same steps as the census sheets — its reflected light and glow on the visible grid, its white, and the share of the brightener’s excitation the ink’s vehicle passes — which reproduce the census sheets exactly at their own loadings. Each tint is the earlier essays’: a Yule–Nielsen mixture with n = 3 of ink and bare sheet, the glow from bare paper in full and from inked paper through the vehicle’s share, a light ink’s coverage matched in Oklab lightness on the sheet without fluorescence, and the residue read by a reader fully adapted to the sheet by Bradford. The census line is the least-squares straight line through each pigment’s three census strengths against their sheets’ glow.

What this does not reach #

A base that stays white. Photo-ageing yellows the base paper as it bleaches the brightener, and that is held fixed here; a yellowing base would move the bleached sheets’ strengths further from the line, and could cancel the shade by which they sit under it.

One stock bleached. Only the premium sheet is bleached. The office sheet, bleached, would pass through the light sheet’s glow, and the same reasoning would set its ink for the light sheet’s — untested.

Uniform bleaching. A real sheet bleaches where light falls on it, and a page’s margins and its centre may age differently; a light ink cannot be set for both.

Still open: whether a yellowing base moves the line #

The bleached sheets sit a shade under the census line, and the likeliest reason is their base paper, held white while the brightener goes. Real photo-ageing yellows the base too, and how far it yellows for a given share of brightener lost depends on the paper’s furnish and the light it is kept under.

The calculation is the premium sheet bleached by a quarter, a half and three quarters with its base’s yellowing raised in step from 0.13 towards the light sheet’s 0.17, each pigment’s cancelling strength read against the census line, and the office ink’s residues read again. The prediction is that the yellowing lifts the bleached strengths onto the line or just above it, within 0.01 at every stage, and that the office ink still keeps every residue within one and a half degrees. If the yellowed sheets move off the line by more than the bleached ones did, the base paper’s colour enters the cancelling strength on its own account, and a printer must read a sheet’s whiteness as well as its glow.

The setting a system passes through #

The habit here is about calibrating something that drifts.

A light ink is a calibration: it is set so that one error vanishes under one condition. The natural setting is the condition at hand — the fresh sheet, on the day of printing — and that setting guarantees the error is zero once and grows in one direction for the rest of the sheet’s life.

The failure mode is to calibrate a drifting system at its starting point. Every later reading is then wrong in the same direction, and by more as time passes. The test is to ask what range the condition will pass through, and to set the calibration where the system will spend the middle of its life — here, at a glow the premium sheet reaches half to three quarters bleached — so that the error changes sign once and stays small on either side.